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akak analogous analytic analytic function arbitrarily arbitrary assume bkak boundary conditions boundary-value problem bounded C2 in 2D calculation characteristic curves characteristic manifolds coefficients completely continuous consider const constants continuously differentiable converges defined denote determined domain 2D eigenvalue estimate existence Figure fixed follows immediately formula Furthermore Gauss theorem given gives grad Green's formula Green's function heat equation Hence Hilbert space Holder-continuously differentiable hyperbolic hypotheses inequality initial conditions initial values initial-value problem integral theorem Let u(x linear Math maximum-minimum principle mean-value property neighborhood normal domain normal form notation obtain operator ordinary differential equations parabolic partial differential equations prescribed Proof radius real numbers right-hand side satisfies Section 4.1 sequence singularity function solution u(x solved sphere suitable suppose transformation ui(x uniform convergence uniformly uniquely unknown functions vanish variables vector field wave equation Weyl's lemma zero